Package jline.solvers.fluid.moments

Class FluidRefineMeanfield

java.lang.Object

jline.solvers.fluid.moments.FluidRefineMeanfield

public class FluidRefineMeanfield
extends Object

Refined mean field correction of a fluid fixed point (Gast, POMACS 2017).

The mean-field fixed point x* is the leading term of an expansion of the true stationary mean in powers of the system size. The next term is obtained by carrying the second moment through the drift: writing A for the Jacobian at x* and B for its Hessian tensor, the correction V solves the linear system

   A*V + (1/2) * sum_{j,k} Sigma_{jk} * d2F/dx_j dx_k = 0
 

with Sigma the stationary covariance from FluidLyapunov. Because Sigma scales with the population, V is the O(1/N) term of the expansion written directly in job counts, so no explicit density rescaling is needed. The Hessian contraction is evaluated WITHOUT EVER FORMING THE TENSOR: writing Sigma = sum_m lam_m*v_m*v_m' by eigendecomposition,

   sum_{jk} Sigma_{jk} d2F/dx_j dx_k = sum_m lam_m * d2F/dv_m^2
 

and each directional second derivative is one central second difference, so the cost is O(rank(Sigma)) drift evaluations rather than O(n^2).

The drift must be TWICE DIFFERENTIABLE for this to mean anything. The first-order closure is only piecewise linear -- its second derivative is zero away from the kink and a delta at it -- so this must be called on the Gaussian-closed drift, i.e. with the sigma2 that the moment closure converged to under options.method='refined'. Passing sigma2 = 0 is rejected rather than silently returning zero.

Port of matlab/src/solvers/FLD/fluid_refine_meanfield.m.

See Also:

• FluidLyapunov
• FluidMomentTerms

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	FluidRefineMeanfield.Result	The correction, plus what it took to compute it.

Method Summary

Modifier and Type	Method	Description
static FluidRefineMeanfield.Result	refine(double[] x, double[] sigma2, Matrix sigma, FluidMomentTerms terms, Matrix[] covblk)	The O(1/N) correction of a fluid fixed point.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

refine

public static FluidRefineMeanfield.Result refine(double[] x,
 double[] sigma2,
 Matrix sigma,
 FluidMomentTerms terms,
 Matrix[] covblk)

The O(1/N) correction of a fluid fixed point.

Parameters:

x - fluid fixed point

sigma2 - per-station population variances defining the smooth drift

sigma - (n x n) stationary covariance

terms - representation from FluidMomentTerms

covblk - per-station covariance blocks closing the DPS share ratio

Returns:

the correction and its diagnostics